On the cycle class map for zero-cycles over local fields

TL;DR

This paper investigates the cycle class map for zero-cycles on smooth projective varieties over local fields, proving injectivity in specific cases and exploring its limitations.

Contribution

It establishes the injectivity of the cycle class map for a broad class of surfaces over local fields and examines its behavior for semistable K3 surfaces.

Findings

Injectivity holds for many surfaces with positive geometric genus over local fields.

The cycle class map is injective for semistable K3 surfaces over C((t)).

Injectivity does not generally hold for surfaces over strictly local fields.

Abstract

We study the Chow group of zero-cycles of smooth projective varieties over local and strictly local fields. We prove in particular the injectivity of the cycle class map to integral l-adic cohomology for a large class of surfaces with positive geometric genus, over local fields of residue characteristic different from l. The same statement holds for semistable K3 surfaces defined over C((t)), but does not hold in general for surfaces over strictly local fields.